Finite Group Databasebeta

← #29692217

Subgroups — #29692217

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic
2#21448567C211yesmaximal, characteristic, complemented
3#21448463C211yesmaximal, characteristic
4#21446770C211yesmaximal, characteristic, complemented
5#21446589C211yesmaximal, characteristic, complemented
6#21446562C211yesmaximal, characteristic, complemented
7#21446547C211yesmaximal, characteristic, complemented
8#21446516C211yesmaximal, characteristic
9#21446502C211yesmaximal, characteristic
10#21446447C211yesmaximal, characteristic
11#21446440C211yesmaximal, characteristic
12#21446405C211yesmaximal, characteristic, complemented
13#21446398C211yesmaximal, characteristic, complemented
14#21445830C211yesmaximal, characteristic
15C61×#1768C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840053C418yescomplemented
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27#16839666—21no—
28#16839666—21no—
29#16839664C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839651—21no—
36#16839651—21no—
37#16839645—21no—
38#16839645—21no—
39#16839628C2211yescharacteristic
40#16839624—21no—
41#16839621—21no—
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#376C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63#14519777—22no—
64#14519777C2×C414yes—
65C4×D244—22no—
66C4×D244C2×C414yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519711—41no—
88#14519709—41no—
89#14519709C2×C414yes—
90#14519709C2×C414yes—
91#14519699—41no—
92#14519699—41no—
93#14519699C2×C414yes—
94#14519699C2×C414yes—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C4⋊C8×C61—22no—
112C4⋊C8×C61C2×C414yes—
113#97×C61—41no—
114C4×C8×C61—41no—
115#95×C61D411yescharacteristic
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119C23×D61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244C22×C412yes—
140C2×C4×D61—24no—
141C2×C4×D61—81no—
142C2×C4×D61—81no—
143C2×Dic122—24no—
144C2×Dic122—81no—
145C2×Dic122—81no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61C22×C412yes—
157C2×C8×C61C22×C412yes—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61C22×C412yes—
163C42×C61#4112yes—
164#11758198—24no—
165#11758198—81no—
166#11758198—81no—
167#11758198—81no—
168#11758198—81no—
169C4⋊Dic61—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C22×C412yes—
184#11758196—81no—
185#11758196—81no—
186#11758196—81no—
187#11758196—81no—
188C4×Dic61—81no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C23×C61C22×D411yescharacteristic
192C22×D61—22no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61C4×D412yes—
216C2×C4×C61C4×D412yes—
217C2×C4×C61C23×C411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—22no—
245D244—22no—
246D244—41no—
247D244—41no—
248C4×D61—84no—
249Dic122—84no—
250C8×C61—22no—
251C8×C61—22no—
252C8×C61—41no—
253C8×C61—41no—
254#22123—611nomaximal, Sylow
255C22×C61C2×C4×D411yescharacteristic
256C22×C61#53611yescharacteristic
257C22×C61#53711yescharacteristic
258C22×C61#57611yescharacteristic
259C22×C61#57711yescharacteristic
260C22×C61#57711yescharacteristic
261C22×C61#57711yescharacteristic
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61#51911yescharacteristic
276C4×C61#52011yescharacteristic
277C4×C61#54111yescharacteristic
278C4×C61#54211yescharacteristic
279Dic61—42no—
280Dic61—42no—
281Dic61—42no—
282Dic61—81no—
283Dic61—81no—
284Dic61—81no—
285Dic61—81no—
286C2×#476—611no—
287#2331—611no—
288#2319—611no—
289#1988—611no—
290#1949—611no—
291#1944—611no—
292#1941—611no—
293#1936—611no—
294#1934—611no—
295#1926—611no—
296#1925—611no—
297#1920—611no—
298#1919—611no—
299#1804—611no—
300#1768D6111yescharacteristic, complemented
301C2×C61#232311yescharacteristic
302C2×C61#294611yescharacteristic
303C2×C61#294911yescharacteristic
304C2×C61#324111yescharacteristic
305C2×C61#324411yescharacteristic
306C2×C61#325711yescharacteristic
307C2×C61#326311yescharacteristic
308D61—82no—
309C2×C4⋊Q8—611no—
310C2×C4×D4—611no—
311#476—618no—
312C2×#107—611no—
313C2×#106—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×C4⋊C8C2×D6111yescharacteristic
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×#97C2×D6111yescharacteristic
323C2×C4×C8C2×D6111yescharacteristic
324#391—1221no—
325#391—1221no—
326#390—611no—
327#390—611no—
328#389—611no—
329#389—611no—
330#389—611no—
331#389—611no—
332#388—1221no—
333#388—1221no—
334#387—1221no—
335#387—1221no—
336#382—611no—
337#381—1221no—
338#380—1221no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—611no—
342#376C2×D6111yescharacteristic
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#336C2×D6111yescharacteristic
352#336C2×D6111yescharacteristic
353C61#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1221no—
355C22×D4—611no—
356C23×C4—1221no—
357C22×C8C22×D6111yescharacteristic
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C4⋊Q8—614no—
362C4⋊Q8—1222no—
363C4×D4—614no—
364C4×D4—1222no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C42—1221no—
383C2×C42—1221no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386C2×C42C22×D6111yescharacteristic
387C2×C42C22×D6111yescharacteristic
388#107—2441no—
389#106—614no—
390#106—614no—
391#106—2441no—
392C4⋊C8—22no—
393C4⋊C8C4×D6114yes—
394#102—614no—
395#102—614no—
396#102—2441no—
397#102—2441no—
398#97—41no—
399C4×C8—41no—
400#95—1221no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95D24411yescharacteristic
410#95D24411yescharacteristic
411#95D24411yescharacteristic
412#95D24411yescharacteristic
413C24—1221no—
414C2×Q8—1224no—
415C2×Q8—4881no—
416C2×Q8—4881no—
417C2×D4—612no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—4881no—
432C22×C4—4881no—
433C22×C4C2×D24411yescharacteristic
434C22×C4C2×D24411yescharacteristic
435C22×C4C2×D24411yescharacteristic
436C22×C4C2×D24411yescharacteristic
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D6112yes—
451C2×C8C2×C4×D6112yes—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1224no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—4881no—
468C4⋊C4—4881no—
469C4⋊C4—4881no—
470C4⋊C4—4881no—
471C22⋊C4—1224no—
472C22⋊C4—4881no—
473C22⋊C4—4881no—
474C22⋊C4—4881no—
475C22⋊C4—4881no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—4881no—
481C42—4881no—
482C42—4881no—
483C42C2×C4×D6112yes—
484C42#1175822212yes—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C22×D24411yescharacteristic
493Q8—4884no—
494D4—1222no—
495D4—1222no—
496D4—2441no—
497D4—2441no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—4884no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C4×D24412yes—
537C2×C4C4×D24412yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4#1451982211yescharacteristic
541C2×C4#1451982411yescharacteristic
542C2×C4#1451982411yescharacteristic
543C2×C4#1451982511yescharacteristic
544C8—22no—
545C8—22no—
546C8—41no—
547C8—41no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×C4×D24411yescharacteristic
556C22#1684033811yescharacteristic
557C22#1684034311yescharacteristic
558C22#1684056711yescharacteristic
559C22#1684057411yescharacteristic
560C22#1684057411yescharacteristic
561C22#1684057411yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—4881no—
569C4—4881no—
570C4—4881no—
571C4—4881no—
572C4—2442no—
573C4—2442no—
574C4—2442no—
575C4#1684023811yescharacteristic
576C4#1684024111yescharacteristic
577C4#1684037011yescharacteristic
578C4#1684037711yescharacteristic
579C2—4882no—
580C2#2144850211yescharacteristic
581C2#2145465311yescharacteristic
582C2#2145468011yescharacteristic
583C2#2145707411yescharacteristic
584C2#2145711911yescharacteristic
585C2#2145723211yescharacteristic
586C2#2145729011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1768C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1768D61
hypocenterC61#22123
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21448567C2semidirect
4. #21446770C2semidirect
5. #21446589C2semidirect
6. #21446562C2semidirect
7. #21446547C2semidirect
12. #21446405C2semidirect
13. #21446398C2semidirect
15. C61×#1768C2semidirect
18. #16840053C4semidirect
300. #1768D61semidirect
353. C61#22123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400532
#214485672
#214484632
#214467702
#214465892
#214465622
#214465472
#214465162
#214465022
#214464472
#214464402
#214464052
#214463982
#214458302
C61×#17682
#2212361